We prove that there exists a non-abelian group structure on the Urysohn universal metric space. More precisely, we introduce a variant of the Graev metric that enables us to construct a free group with countably many generators equipped with a two-sided invariant metric that is isometric to the rational Urysohn space. We list several related open problems.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm228-3-3,
author = {Michal Doucha},
title = {Non-abelian group structure on the Urysohn universal space},
journal = {Fundamenta Mathematicae},
volume = {228},
year = {2015},
pages = {251-263},
zbl = {1312.22001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm228-3-3}
}
Michal Doucha. Non-abelian group structure on the Urysohn universal space. Fundamenta Mathematicae, Tome 228 (2015) pp. 251-263. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm228-3-3/